Question

the national average for the math portion of the college boards SAT is 521 with a standard deviation of 84 and the median is 521. What is the shape of the SAT math score distribution? Why? and What rule (empirical or Chebyshev's) predicts that how many percent of the SAT math score data should lie between a 353 and 689, which are +/-(number) standard deviation away from the mean?

Answer 1

The SAT math score has the shape of a standard bell curve or normal distribution. This is because the median and the mean has the same value, therefore, the data is centered and distribute around the mean. The farther we go from the mean, the less probable is to find data with those values.

Now, considering that we have established that the data has a normal distribution, we can use the empirical rule to predict how many percent of the SAT math score data should lie between a 353 and 689. We calculate how many standard deviatioms the maximum and minimum value of the range are from the mean:

$\frac{689 - 521}{84} = 2$

$\frac{521 - 353}{84} = 2$

This would mean that we need to calculate the percentage of the SAT math score data that would lie within 2 standard deviations from the mean. The empirical rule says that in a normal distribution 68.27% of the data will lie within 1 standard deviations from the mean, 95% will lie within 2 standard deviations and 99.7 will lie within 3 standard deviations. Having calculated that our range is 2 standard deviations from the mean, then 95% of the students will get a SAT math score between 353 and 689.

In summary, the SAT math score has a normal distribution or standard bell shape, and 95% of the SAT math scores will be between 353 and 689, or 2 standard deviations from the mean.